2 IMPLICIT FUNCTIONS 275

Show that if two continuous mappings «, v of I into G are such that pri(«(/)) —
priWO) = y(0 for each t e I, and if they are equal for one value of t e I, then u = v
(use a similar method).

(d) Under the same assumptions as in (c), let <p be a continuous mapping of I x J
into A, where J = [c, d\ c: R. Let t? be a continuous mapping of J into G such that
priWf)) = <p(<*> f) for £eJ; and for each f ej, let w^ be the unique continuous
mapping of I into G such that pri(w^O)) = <?(/,£) for / e I and u^a) = v(g). Show that
the mapping (t, £) -> #,«(/) is continuous in I x J. (Given £ e J, there is a number r > 0
such that for any t e I, the intersection Vt of T and of the closed ball in E x F, of
center u^(t) and radius r, is contained in G and such that pri is a homeomorphism of
V, onto the closed ball in E of center y(t) and radius r. If L = uff), let M be the
supremum of ||(D2/Gc,j>))~1 ° (£>i/(*» y))\\ for all points (jt,j)eG at a distance
^r of L. Let e > 0 be such that e < r/4 and eM < r/4. Show that if 8 is such that
the relation 1£ — £| ^8 implies \\<p(t, £) — <p(f, £)||< « for /el, then the relation
|£ — £| < 8 implies ||w$(/) — u;(t) \\ ^ r/4 for / e I; prove this by considering the l.u.b. of
the / e I for which the inequality holds, and using (10.2.1).)

(e) Conclude from (d) that if the loop y defined in I = [a, b] is loop homotopic to a
point in A, then any continuous mapping u of I into G such that pri («(/)) = y(/)
for / E I is such that u(b) = u(a). In particular, if A is simply connected (i.e., if any
loop in A is homotopic to a point in A), then pri is a homeomorphism ofG onto A,
i.e. there exists a unique continuously differentiable mapping g of A into F such that
f(x, g(x)) = 0 in A and that (x, g(x)) belongs to G for at least one x e A (cf. (16.28.7),)

7. With the notations of Problem 6, show that the condition prA(G) = A is satisfied
for every connected component G of F in each of the following cases:

(0 /(*» y) =/iGO —/2<*, y\ and there exist numbers R>0, A;>0,/z>0 and a
positive continuous function ;c->H(x) in A such that for ||^|| ** R, H/iO>)|| ^ \\y\f
and ||/a(*,7)11 <H(x)|M|*-*.

(2) F = C, E is a vector space over C, /(#, y) = ey — g(x)9 where g is analytic in A
and#(x) ^ 0 in A (this last condition already ensures that pri(F) = A; observe that
f(x, y) = f(x, y') implies that y' — y is a multiple of 2m, hence for any x € A there is
an open ball U of center x contained in A such that for any connected component
V of prpO-J) n F, pri is a homeomorphism of V onto U; if x is a cluster point of
pri(G), G must have a common point with one of these components V, hence
contains V).

8. (a) If/is a complex valued entire function in Cp, such that/(jc) ^ 0 for every x e Cp,
show that there exists a complex valued entire function g in C" such that f(x) = eaM
(use Problem 7).

(b) Let/ be an arbitrary complex valued entire function in C, which is not identically
0; there is a finite or infinite sequence (an) (with n ^ 1) of complex numbers (which may
be empty) such that \an\ < \an+i\,f(aa) = 0 and for every ceC such that/(c) = 0, the
number of indices n for which an = c is equal to the order o>(c;/); when the sequence
(an) is infinite, lim \an = -foo (9.1.5). Show (with the notations of Section 9.12,

n-»oo

Problem 1) that there exists an entire function # such that

f(z) = eg(z) IIE I —, /z — 1 J, (" Weierstrass decomposition ").

9. Let A and B be two open neighborhoods of 0 in E — Cp, A being connected; let (x, y) ->
U(x,y) be an analytic mapping of A x B into ^f(E; E) (identified to the space of
p x p matrices with complex elements). there is a connected component G of T such